In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense.
The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.
Statement of the Minkowski-Steiner formula
Let \( n\geq 2 \), and let \( {\displaystyle A\subsetneq \mathbb {R} ^{n}} \) be a compact set. Let \( {\displaystyle \mu (A)} \) denote the Lebesgue measure (volume) of A A. Define the quantity \( {\displaystyle \lambda (\partial A)} \) by the Minkowski–Steiner formula
\( {\displaystyle \lambda (\partial A):=\liminf _{\delta \to 0}{\frac {\mu \left(A+{\overline {B_{\delta }}}\right)-\mu (A)}{\delta }},} \)
where
\( {\displaystyle {\overline {B_{\delta }}}:=\left\{x=(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}\left||x|:={\sqrt {x_{1}^{2}+\dots +x_{n}^{2}}}\leq \delta \right.\right\}} \)
denotes the closed ball of radius \( \delta >0 \), and
\( {\displaystyle A+{\overline {B_{\delta }}}:=\left\{a+b\in \mathbb {R} ^{n}\left|a\in A,b\in {\overline {B_{\delta }}}\right.\right\}}
is the Minkowski sum of A and \( {\displaystyle {\overline {B_{\delta }}}} \), so that
\( {\displaystyle A+{\overline {B_{\delta }}}=\left\{x\in \mathbb {R} ^{n}{\mathrel {|}}\ {\mathopen {|}}x-a{\mathclose {|}}\leq \delta {\mbox{ for some }}a\in A\right\}.} \)
Remarks
Surface measure
For "sufficiently regular" sets A, the quantity \( {\displaystyle \lambda (\partial A)} \) does indeed correspond with the (n-1)-dimensional measure of the boundary \( \partial A \) of A. See Federer (1969) for a full treatment of this problem.
Convex sets
When the set A is a convex set, the lim-inf above is a true limit, and one can show that
\( {\displaystyle \mu \left(A+{\overline {B_{\delta }}}\right)=\mu (A)+\lambda (\partial A)\delta +\sum _{i=2}^{n-1}\lambda _{i}(A)\delta ^{i}+\omega _{n}\delta ^{n},} \)
where the λ i {\displaystyle \lambda _{i}} \lambda_{i} are some continuous functions of A (see quermassintegrals) and \( \omega _{n} \) denotes the measure (volume) of the unit ball in \( } \mathbb {R} ^{n} \):
\( {\displaystyle \omega _{n}={\frac {2\pi ^{n/2}}{n\Gamma (n/2)}},} \)
where Γ {\displaystyle \Gamma } \Gamma denotes the Gamma function.
Example: volume and surface area of a ball
Taking \( {\displaystyle A={\overline {B_{R}}}}\) gives the following well-known formula for the surface area of the sphere of radius R, \( {\displaystyle S_{R}:=\partial B_{R}} \):
\( {\displaystyle \lambda (S_{R})=\lim _{\delta \to 0}{\frac {\mu \left({\overline {B_{R}}}+{\overline {B_{\delta }}}\right)-\mu \left({\overline {B_{R}}}\right)}{\delta }}} \)
\( {\displaystyle =\lim _{\delta \to 0}{\frac {[(R+\delta )^{n}-R^{n}]\omega _{n}}{\delta }}} \)
\( {\displaystyle =nR^{n-1}\omega _{n},} \)
where \( \omega _{n} \) is as above.
References
Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2.
Federer, Herbert (1969). Geometric Measure Theory. New-York: Springer-Verlag.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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